Collimation of high current fast electrons in denseplasmas with a tightly focused precursor intenselaser pulse

H Xu1,2,†, X H Yang3,4,∗, Z M Sheng2,5, P McKenna5, Y Y Ma3,2,H B Zhuo3,2, Y Yin3, C Ren4, and J Zhang2

1College of Computing Science, National University of Defense Technology, Changsha410073, P. R. China2IFSA Collaborative Innovation Center, Shanghai Jiao Tong University, Shanghai200240, P. R. China3Department of Physics, National University of Defense Technology, Changsha410073, P. R. China4Department of Mechanical Engineering, University of Rochester, Rochester, NewYork 14627, USA5Department of Physics, SUPA, University of Strathclyde, Glasgow G4 0NG, UK

E-mail: xuhan−email@aliyun.com

E-mail: xhyang@nudt.edu.cn

Abstract. High current fast electrons at the megaampere level provide a uniqueway to generate high energy density states of matter, which are related to manyapplications. However, the large divergence angle of the fast electrons typically over50 degrees is a significant disadvantage for applications. The guiding effect by theself-generated azimuthal magnetic fields of the fast electron current is found to be verylimited due to their cone-shaped spatial structure. In this work, we present a newunderstanding on the collimation conditions of fast electrons under such a magneticfield structure. It is shown that the transverse peak position of the magnetic field layerplays a crucial role to collimate the fast electrons rather than its magnitude. Basedupon this, a new two-pulse collimating scheme is proposed, where a guiding precursorpulse is adopted to form proper azimuthal magnetic fields and a main pulse is forfast electron generation as usual. The present scheme can be implemented relativelyeasily with the precursor lasers at the 10 TW level with a duration of two hundredfemtoseconds, with which the divergence angle of fast electrons driven by the mainpulse can be confined within a few degrees. Our scheme can find practical applicationsin high energy density science.

PACS numbers: 52.25.Fi, 52.57.Kk, 52.65.Ww

Submitted to: Nuclear Fusion

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1. Introduction

The ultra-short high power lasers based upon the CPA technology [1] have enabledbroad applications from fundamental science to industry and medicine. In particular,such lasers push nonlinear optics to the relativistic regime [2], which can drive mattersinto high energy density states [3] with many applications, such as electron andion acceleration [4, 5], compact x-ray [6, 7], neutron sources and isotope production[8–11], the fast ignition approach to inertial confinement fusion [12–14], and laboratoryastrophysics [15], etc. Many of these applications mentioned above are associated withthe high current relativistic electrons produced during the intense laser interaction withsolid targets. Even though the efficiency of energy transfer from relativistic intenselasers to hot electrons can be well above 10% [16], the hot electrons often are producedwith large divergence angle typically over 50 degrees [17], which make it difficult toachieve collimated propagation in solid target. Collimated propagation of fast electronsis crucial for these applications. So far this has not yet been completely solved.

The penetration depth and energy deposition of fast electrons in targets stronglydepend on the material properties (like electrical resistivity) and fast electron beamproperties. However, the fast electrons from laser-solid interaction typically have broadangular distributions due to the beam transport in dense plasma [18–20], the lasertransverse ponderomotive force [21] or the reflection of the laser light [22] in preplasmas.Generally, the divergence angle increases with the laser intensity [23]. On the otherhand, the resistive magnetic fields are self-generated in the solid target, which areazimuthal and hence produce some collimating effect on the electron beam [24,25]. Withthis, the propagation characteristic of fast electrons can be modulated by using somespecially designed targets, such as targets consisted of alternating layers of differentZ materials [26, 27] and low-Z targets doped with high-Z elements [28]. It is alsoreported that the resistivity at low temperature (e.g., several eV), which also relieson the material microstructure, has a significant effect on the resistive magnetic fieldand thus the propagation of the fast electrons [29, 30]. The collimating propagation offast electrons can also be achieved by optimizing the parameters of laser pulses, e.g.,a two-pulse injecting scheme, named guiding pulse and main pulse, was introducedtheoretically [31] and later implemented in experiments [32, 33]. Furthermore, multiplelaser pulses with gradually increasing intensity can also enhance the collimation of fastelectrons [34].

The collimation criterion of fast electrons by the self-generated magnetic field wasfirstly proposed by Bell and Kingham [35] and extended by Robinson et al. [31] andCurcio et al. [36], in which the spatial transverse extent L⊥ of the magnetic field (normalto the laser propagation direction) is assumed to be equal to the focal spot size andthe spatial longitudinal extent L∥ is assumed to be infinite. Therefore the criterion isintrinsically based on a one-dimensional cylindrical structure of the magnetic field. Inreality, however, a cone-shaped magnetic field structure distributed at the peripheryof the laser focal spot with a tilt angle is ubiquitously observed in many numerical

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simulations and experiments [24–32, 35, 37–39]. The collimation criterion with such amagnetic field structure has not been considered and understood yet. In recent years,there has been significant interest to collimate fast electrons by use of high magneticfields [40–43]. The magnetic fields can be as high as 1 kT when nanosecond drivinglasers with energy at kJ level are applied [42, 44]. This has a high demand for thedriving lasers and is not easily accessible for many users.

In this paper, we investigate the conditions for fast electron collimation in thepresence of a typical cone-shaped magnetic field structure driven by an ultrashort intenselaser pulse, which can have a lower peak power than that used to produce fast electrons.Firstly, the magnetic field structure and its dependence on the laser parameters arestudied by using a “rigid-beam” model, i.e., the fast electron current density is fixed [37].It is found that the magnetic field structure has a specific position and a tilt angle. Thetransverse peak of the magnetic field layer moves to outer radii of the laser focal spotand its tilt angle approaches the initial divergence angle of fast electrons as the laserintensity increases. All these features prevent the fast electrons from being collimatedby the self-generated magnetic field. To recover effective collimation, a new two-pulsecollimating scheme is proposed, in which the guiding pulse has a much smaller focalspot than that of the main pulse to produce the suitable magnetic field structure. Eventhough the transverse peak position of the cone-shaped magnetic field generated bythe guiding pulse is still larger than the focal spot radius of the guiding pulse, it issmaller than the focal spot radius of the main pulse and thus can effectively collimatethe fast electrons generated by the main pulse. Finally, three-dimensional (3D) hybridparticle-in-cell (PIC)/fluid simulations are employed to validate this two-pulse scheme.It is shown that the divergence angle of the fast electron beam can be reduced from 25

to 7 (half-width at half-maximum: HWHM) by using a guiding pulse of 10 TW leveltable-top laser system.

2. Theoretical model for cone-shaped magnetic fields driven by intense laser

High current fast electrons at the megaampere level can be driven by tens of terawattsintense laser pulse irradiating a solid target (see Fig. 2a for the schematic of the laser-target interaction), which can induce megagauss magnetic fields to confine the fastelectron propagation [37, 38]. The magnetic field structure depends on the laser pulseparameters, e.g., laser intensity and pulse duration, and can be obtained by using the“rigid-beam” model. The driven current density are assumed to have a Gaussian profile(having only z component Jf = Jf ez), varing with beam radius r and propagating withthe speed of light c, that is

Jf (r, z, τ) = J0R0

Rz

exp

(−2r2

R2z

)f(τ), (1)

where f(τ) is the temporal profile, τ = t − z/c is the pulse duration (time passingthrough a given point). J0 ≡ Jf |r=0,z=0 is the current density at z = 0 plane that is

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given by the conservation of energy flux βeI0 = J0⟨E⟩, where β is absorption coefficient(usually between 0.2-0.4), ⟨E⟩ is the averaged energy of fast electrons determined byWilks’ scaling law [45] ⟨E⟩/(mec

2) =√

1 + I0λ20/1.38× 1018 − 1, I0 is the laser peak

intensity in units of W/cm2, λ0 is the wavelength in units of µm, Rz = R0 + z · tan(θ)is the spot size of fast electron beam at z, R0 is the spot radius at the incident plane(z = 0), and θ is the half-divergence of the fast electron beam.

For a high temperature plasma, e.g., Te > 50 eV for (CH2), the Spitzer-Härmresistivity [46] η(Te) = η0(Te/T0)

−3/2 is a good approximation, where T0 and η0 arethe initial temperature and resistivity, respectively. On the assumption of the balancebetween the fast electron current and the cold return current, and assuming f(τ) = 1

during the laser pulse period 0 ≤ τ ≤ τL, the azimuthal component of magnetic fieldbecomes [47]

Bϕ =4r

R2z

CvT0

Jf

[1− 1 + 0.5τ ′

(1 + 2.5τ ′)0.6

], (2)

where Cv = 32Zni is the electron specific heat capacity, ni is the ion density, Z is the

averaged ionization degree of the material. Both ni and Z are assumed to be constantshere. The normalized variable τ ′ = τ/τT measures the heating rate of the target [37],where τT ≡ CvT0/η0J

2f is the typical heating time to temperature T0.

The transverse peak position of the magnetic field Bϕ(r, z) lies at ∂Bϕ/∂r = 0. Bytaking the derivation of Eq. (2) and some operations, the transverse peak position rmof the magnetic field is approximately govern by

τ ′(rm, z) ≃1

3. (3)

Substituting Eq. (1) and the definition of τ ′ into Eq. (3), then taking natural logarithmon both sides of the equation, the peak position rm explicitly reads

rm(z)

Rz

=1

2

√ln[3τ ′0,0(1−

z

ct)]+ 2 ln(R0/Rz), (4)

where τ ′0,0 ≡ τ ′|r=0,z=0 = CvT0t/η0J20 . Substituting Eqs. (1) and (4) into Eq. (2), the

peak magnitude of Bϕ at plane z = 0 reads

Bϕ,m = 0.66√

τ ′0,0 ln(3τ′0,0) · CvT0/R0J0. (5)

The relation of τ ′0,0 with laser intensity is given by [37]

τ ′0,0 = 8.48× 1019η5/30 β2I0τL

(Z ln Λ)2/3neλ20

, (6)

where β = 0.25, ln Λ is the Coulomb logarithm, η0 = 2.3µΩ ·m and Z ln Λ = 7.7 is usedhere for CH2 [48]. The normalized heat rate is about τ ′0,0 ≃ 190 for I0 = 1.0×1019W/cm2

and τL = 1ps. A strong Bϕ,m ∝√I0τL ln(3I0τL) can be achieved via the increase of the

pulse intensity I0 or pulse duration τL.

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Figure 1. Distributions of the azimuthal magnetic field Bϕ at t = 1.0 ps obtainedfrom a numeric solution of the “rigid-beam” model (a) and a 3D hybrid PIC/fluidsimulation (b). The black dashed line and white solid line in (a) are the location ofτ ′(r, z) = 1/3 and the fast electron spot radius Rz at z, respectively. The transversepeak position rm of the magnetic field at z = 0 plane (c) and the averaged tilt angle α

(d) near the focal spot. The “rigid-beam” model has the identical parameters to thatin the hybrid simulation described in the text.

The tilt angle α of the magnetic field layer can be estimated by tan(α) ≈[rm(z1/2) − rm(0)]/z1/2, where z1/2 is the longitudinal position for Bϕ,m/2, and rm(0)

and rm(z1/2) are the transverse position of Bϕ,m and Bϕ,m/2, respectively. ApplyingEqs. (1), (2), and (4), and since z/ct ≪ 1 near the focal spot, the tilt angle can beapproximated as

tan(α) ≈ tan(θ)

√A

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1− 2x0√x0 − x0

, (7)

where x0 = [2.39 − A +√A2 − 0.78A+ 5.7]/8 and A ≡ ln(3τ ′0,0). For a high intensity

and/or a long duration of laser pulse, e.g., I0 > 1×1019W/cm2 and τL = 1.0 ps, it showsτ ′0,0 ≫ 1 and x0 → 1/4, which leads to rm(0) =

√A2R0 and tan(α) =

√A2

tan(θ), i.e., wegenerally have

rm(0) > R0, α > θ. (8)

This means that the self-generated magnetic field layer is located outside of the laserfocal spot. Due to the transverse Gaussian profile of the main laser pulse, only 4.5%of fast electrons are located out of the focal spot radius (r > R0) and 0.6% of fastelectrons are located out of the magnetic field layer (r > 1.36R0, see Fig. 1c). Thus,most of the fast electrons have no chance to enter into the magnetic field area and tobe collimated by the field, although the magnitude of the peak magnetic field is stillstrong enough (up to 500T) to collimate the fast electrons according to the estimationof Robinson et al. [31].

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Figure 1 shows the structure of the self-generated magnetic field from “rigid-beam”model (Fig. 1a) and a 3D hybrid simulation (Fig. 1b). In both cases, we set the peakintensity I0 = 4× 1019W/cm2, spot radius R0 = 12.75µm (corresponding to full-widthat half-maximum (FWHM) of 15µm), pulse duration 1.0 ps, and the half-divergenceangle 30. The target is polyethylene (CH2) with a uniform density of 1.0 g/cm3. Aninitial electron temperature of 48.3 eV is set to keep the initial resistivity 2.3µΩ ·m closeto that used in reference [48]. Figure 1a and 1b show a similar magnetic field structure,both appearing within a narrow band at the periphery of the focal spot. The tilt angle(32.6) in Fig. 1a is larger than that of the hybrid simulation in Fig. 1b (27). Theblack dashed line in Fig. 1a is the location of the peak position rm of Bϕ defined by Eq.(3), which shows a good agreement with the numerical result.

The dependence of rm and the tilt angle α on laser intensities are shown in Figs.1c and 1d. It is seen that both the rm and α increase with laser intensity, and theanalytical results (Eqs. (4) and (7)) are in good agreement with the numerical results ofthe “rigid-beam” model. Note that the tilt angle α from the hybrid simulations is muchsmaller than that from the “rigid beam” model. This is due to the feedback missing ofthe collimation effect of the fast electrons in the “rigid beam” model, especially at lowerlaser intensities.

3. Conditions for fast electron collimation by the cone-shaped magneticfields

To investigate the confinement of fast electrons by the self-generated magnetic field witha cone-shaped distribution, a simplified structure of the self-generated magnetic field isadopted as shown in Figs. 2. The magnetic flux density B0 is uniformly distributed withspatial length of L, thickness of d, and tile angle of α. It covers the main features of themagnetic field obtained from the “rigid-beam” model (Fig. 1a) and hybrid simulations(Fig. 1b). We consider test fast electrons, which are injected from the left boundarywith an initial angle of θ and a transverse offset R to the the magnetic field layer. Thetrajectory of the fast electron (red lines with arrows) is a part of circle (grey dashed-linein Fig. 2a) in a uniformly perpendicular magnetic field, where r0 = γ0mev0/eBϕ is theLarmor radius, and v0, e,me, γ0 are the fast electron velocity, charge, mass, and Lorentzfactor, respectively.

Note that there are only four cases that the fast electron can be confined by themagnetic field. For the first two cases, the injecting angles of the fast electrons aregreater than the tilt angle of the magnetic field (i.e, α < θ) but with a different transverseoffsets R are shown in Figs. 2a and 2c. For the other two cases, the fast electrons withα > θ are shown in Figs. 2b and 2d. By considering the geometrical relationshipbetween the fast electron trajectory and magnetic field structure, one can obtain the

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Figure 2. (Color online) Sketch of fast electrons deflected/collimated by an azimuthalmagnetic field Bϕ. The magnetic field is azimuthal and points outward, with a tilt angleα, spatial length of L, and thickness of d. The laser incidents from the left boundary(thick dashed-dot line in (a)) and produces a sampled fast electron, which has an initialdivergence angle of θ and an initial transverse offset R to the lower/upper edge of themagnetic layer. The trajectory of fast electron is marked by a red line with an arrowand r0 is the Larmor radius of the fast electron.

fast electron collimating conditions for the above four cases,

Case a :

d > d1,

L > 2r0 sin(θ/2) + L2 ,(8a)

Case b :

d > d2,

L > L1 + L2,(8b)

Case c :

d > d2 +R cos(α),

L > L1,

R > d1/ cos(α),

(8c)

Case d :

d > d2 +R cos(α),

L > L1.(8d)

where d1 = r0[1 − cos(θ/2)], d2 = r0[cos(α − θ) − cos(α)], L1 = r0 sin(θ)/ cos(α), andL2 = R cos(θ)/ sin(|θ− α|). These conditions define the required magnetic field sizes toachieve electron collimation. One notes that Eq. (8a) reduces to the result of Robinsonet al. when α = 0 and R = 0 [31]. When the tilt angle of magnetic field approaches theelectron divergence, i.e., α → θ, then L2 → ∞, thus L in case (a) and case (b) becomevery large, which leads to that only cases (c) and (d) are effective to confine the fastelectrons. The fast electrons would be collimated by the magnetic field if only they can

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satisfy one of Eqs. (8a)-(8d). Once the initial relative position r (in Fig. 2) of the fastelectrons and the structure of magnetic field are given, the fraction of collimated fastelectrons can be estimated.

For the more realistic case, the collimation effect of fast electrons by the cone-shaped magnetic field needs to be evaluated by numerical simulation. We follow thetrajectories of a total of 106 fast electrons, sampled with different r according to thetransverse Gaussian distribution function exp(−2r2/R2

0), which are bent in the self-generated magnetic field obtained numerically from the above “rigid-beam” model. Theresults are shown as the “circle-black-dashed-line” and “circle-black-solid-line” in Fig.3 for two different pulse durations. It can be seen that the fraction of collimatedfast electrons decreases rapidly with the increase of the laser intensity. This can beattributed to two reasons: Firstly, the energy of the fast electrons increases with laserintensity according to Wilk’s scaling law, indicating that a stronger magnetic fieldto be required to collimate the fast electrons; Secondly, both the transverse position(rm/Rz in Eq. (4) and Fig. 1c) and the tilt angle (α in Eq. (7) and Fig. 1d) of peakmagnetic field increase, leading to that less fast electrons can be deflected. Actually,fast electrons are hardly collimated by the self-generated magnetic field alone as laserintensity I0 > 3× 1019W/cm2 (i.e., less than 1%). All these suggest that fast electronscannot be collimated by the self-generated magnetic fields themselves.

In order to confine the fast electrons effectively for a high laser intensity, we proposea two-pulse scheme by use of a guiding pulse and a main pulse, in which the guiding laserpulse with a much smaller focal spot than the main pulse is first incident to generatea cone-shaped magnetic field structure and then the main pulse is injected. The pre-generated magnetic field structure has a size larger than the focal spot of guiding pulsebut still smaller than the spot of the main pulse, which enable to collimate the fastelectrons generated by the main pulse. With this scheme, the fast electrons with incidentangle θ < α can be collimated under the cases shown in Figs. 2b and 2d, while the fastelectrons with incident angle θ > α will be collimated under the cases shown in Figs.2a and 2c. To demonstrate the effectiveness of this scheme, we carry out numericalsimulations based upon the “rigid-beam” model as shown above. Similarly, a total of106 test fast electrons are checked for this scheme. Figure 3 shows the ratio of collimatedfast electrons against the total number of injected electrons under different focal spotsizes and pulse durations of the guiding laser pulse. One can seen that, by applying aguiding pulse with a smaller focal spot radii (R0/2 and R0/3, respectively), the fractionof collimated fast electrons increase drastically for a much stronger laser intensity. Inaddition, the fraction of collimated fast electrons can be enhanced by increasing theduration of the guiding pulses. Furthermore, such collimation becomes more efficient asthe spot radius decreases from R0/2 to R0/3. It is shown that the collimated electronnumber is increased by ∼16 times for the case with a laser intensity of 3× 1019W/cm2

and a guiding pulse with a spot radius of R0/3 and duration of 1 ps compared to thatof without the guiding pulse.

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Figure 3. The fraction of collimated fast electrons for cases with/without precursorguiding pulses. ncoll and ntot are the numbers of collimated and total injected fastelectrons, respectively. τG and τM are the pulse durations of the guiding pulse andmain pulse, respectively. The “circle-black-dashed-line” and “circle-black-solid-line”are the results without the guiding pulse at t = 0.5 ps and 1.0 ps, respectively. The“diamond-blue-line” and “pentagram-red-line” correspond to the cases with the guidingpulses with spot radii of R0/2 and R0/3, respectively, where R0 = 12.75µm is the spotradius of the main pulse. Here I40 means the intensity of guiding pulse is 40 × 1018

W/cm2.

4. 3D HYBRID SIMULATIONS

In order to validate the two-pulse scheme further, some 3D hybrid PIC/fluid simulationsare carried out by using the code HEETS [28], which treats the fast electrons as afully dynamical method including collisions (i.e., explicit PIC), while the backgroundelectron-ion plasmas are described by a reduced two-fluid model. The simulation box is200× 200× 200µm3 with grid of ∆x = ∆y = ∆z = 1µm. Both the guiding pulse andthe main pulse are modeled by setting identical constant intensity I0 = 4×1019W/cm2,and energy efficiency from laser-to-fast electrons of 25%. The main pulse havinga duration of 2 ps and a transverse Gaussian radial profile ∼ exp(−2r2/R2

0) withR0 = 12.75µm, following the guiding pulse of a duration of τguid = 0.2 ps, Gaussianradial profile of Rguid = R0/2 or R0/3, and a time delay of 0.2 ps for the major simulation.The angular distribution of fast electrons is specified to be g(θ) = cosM(δθ), whereM = 6, δ = cos−1(2−1/M)/θ0, θ0 = π/6 is HWHM divergence angle [23] for boththe guiding and main pulse. The energy distribution of fast electrons are assumedto be h(E) = exp(−E/⟨E⟩)/⟨E⟩, where the averaged energy ⟨E⟩ is given by Wilks’scaling law [45]. The target is polystyrene with a uniform density of 1.0 g/cm−3 and aninitial temperature 1 eV. The target resistivity is η = (1/ηmax + 1/ηSP )

−1 [48], where

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ηmax = 2.3µΩ · m is the maximum of the resistivity, ηSP = 7.7 × 10−4T−3/2e is the

Spitzer-Härm resistivity [46]. The average ionization degree of the target is determinedby the Thomas-Fermi model [49].

Figure 4 shows the isosurface of fast electron density (sampled at 4 × 1025m−3),the corresponding density distribution at the target rear surface, and self-generatedmagnetic fields Bx and By components for the cases with/without a guiding pulse att = 1.6 ps. Note t = 0 is corresponding to the time at which the leading edge of the mainpulse enters the left boundary of the simulation box. For the case without a guidingpulse (see Fig. 4a), it can be seen that the fast electrons are divergent in the targetwith an average divergence angle about 25, which is smaller than the initial angle 30

due to the weak confinement effect of the self-generated magnetic field. By employinga precursor guiding pulse, the divergence angle of fast electrons is reduced significantly,e.g., 15 for a guiding pulse with the spot radius of R0/2 (see Fig. 4b) and only 7

for the case of R0/3 (see Fig. 4c). One can see that, in Fig.4c, a small handlebar-likemagnetic field structure (peak magnitude about 400 T) appears around the focus center.Though the magnitude of this field is weaker than that due to the main pulse (about580 T), it plays an important role to confine the fast electrons due to the fact that it islocated in the region of fast electron propagation. When the spot radius of the guidingpulse becomes larger, this handlebar-like magnetic field moves outward and merges withthe magnetic field of the main pulse, as shown in Fig. 4b, so the confinement becomesweaker.

To investigate the influence of the parameters of the guiding pulse on the fastelectron collimation, more simulations are carried out. The fast electron spot radiusRout at the rear surface of the target and averaged target temperature Tavg within Rout

are recorded for different spot radii (Fig. 5a), different pulse durations (Fig. 5b) of theguiding pulses, and different time-delay between the guiding pulse and main pulse (Fig.5c). The averaged temperature of the target is relevant to the fast electron energy fluxpenetrating through the rear surface, which can directly show the collimation effect.It is shown that the guiding pulse has a optimum spot radius around 5µm (FWHM).Further decreasing the spot radius will lead to the increase of number of fast electronsmissing the confinement of the magnetic field. In the “rigid-beam” model, the peakmagnetic field is dependent on the product of pulse duration τ and intensity I0, i.e.,Bϕ,m ∝

√I0τ ln(3I0τ) (see Eqs. (5)), which indicates that a longer pulse duration will

induce a stronger magnetic field and so a better confinement, just as shown in Fig. 5b.One can note that, in Fig. 5b, a pulse duration of 200 fs (energy 2.27 Joule with peakpower of 11.35 TW) drastically reduces the fast electron divergence. In that case, thefast electron spot radius at the rear surface of the target is about 35µm comparingto 105µm for the case without a guiding pulse. Fig. 5c shows that the influence oftime-delay between the guiding pulse and main pulse (at least within 2 ps) is very weak.This can be attributed the long lifetime of magnetic field structure. The characteristicdiffusion length of the magnetic field is given by Ld = (ητd/µ0)

1/2 [50,51], where η is theresistivity of target, µ0 is the permeability of free space, τd is the time delay between the

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Figure 4. Isosurfaces (sampled at 4 × 1024m−3) and log10 of fast electron numberdensity on the rear surface z = 200µm at t=1.6ps from 3D hybrid simulations for thecases without guiding pulse (a), guiding pulses with spot radii of R0/2 (b) and R0/3

(c) at t = 1.6 ps, respectively. Distributions of the self-generated magnetic field Bx atx = 0 plane and By at y = 0 plane are also presented. The density and magnetic fieldare in units of m−3 and tesla, respectively. t = 0 corresponds to the leading edge ofthe main pulse crosses the left boundary of the simulation box. Other parameters areas follow: both the intensities of main and guiding pulses are I0 = 4×1019W/cm2, theduration of main and guiding pulses are 2 ps and 0.2 ps, respectively, the focal spotradii are R0 = 12.75µm (R0,FWHM = 15µm) for the main pulse, and R0/2 (b) andR0/3 (c) for the guiding pulses, the delay time between guiding and main pulses is 0.2ps.

two pulses. For materials that have been heated and ionized, with maximum resistivityaround 10−6Ω·m, one can find that Ld is very small for the picoseconds time delay,e.g., Ld ∼ 0.4µm and ∼ 1.26µm, respectively, for τd=0.2 ps and 2 ps. That is, afterthe magnetic field is generated by the guiding pulse, the magnitude and structure ofmagnetic field will not have obvious variance in such short time delay. It indicates thatour scheme should be robust enough to be implemented in future experiments. Suchtwo pulses can be originated from the same oscillator and be temporally synchronizedusing interferometry techniques. One can use two different off-axis parabolic mirrors,one for each laser pulse, to vary the focal spot size of the pulses.

5. CONCLUSION

In summary, we have investigated the magnetic field structure produced during the fastelectron transport in dense plasma, where the fast electrons are generated by relativisticlaser interaction with solid targets. The magnetic field structure typically has a cone-shape with an open angle, which increases with the driving laser intensity. It is alsolocated outside the focal spot of the driving laser pulse. It is shown that fast electronsgenerated by high laser intensity are hardly confined by this self-generated magnetic field(e.g., less than 1% of the total fast electrons can be collimated as I0 > 3× 1019W/cm2)due to such magnetic field structure. The criteria for collimating fast electron beam

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Figure 5. The spot radius of fast electrons Rout (red y-axis) and the averaged electrontemperature Tavg within Rout (blue y-axis) at the rear surface, for (a) different focalspot radii of the guiding pulse with duration τG = 0.2 ps, (b) different guiding pulseduration with focal spot radius RG = R0/3, and (c) different delay time between theguiding and main pulses with τG = 0.2 ps, RG = R0/3. Other parameters are samewith that in Fig. 4.

transport in dense plasmas in the presence of a cone-shaped magnetic field structureare presented. Based upon these, a two-pulse collimating scheme is proposed, in whicha precursor guiding pulse with a focal spot smaller than the main pulse is appliedto generate a favorable magnetic field layer. The validity and robustness of the two-pulse collimating scheme are verified by 3D hybrid simulations. It is shown that sucha magnetic field structure driven by a 10 TW laser can already produce significantcollimating effect, where the divergence angle of fast electron beam can be decreasedfrom 25 to 7 by employing a guiding pulse with a spot radius of R0/3. The timedelay between the guiding laser pulse and the main laser pulse is relatively flexible (atleast within 2 ps considered here). Thus, even a relatively low-cost compact tabletopfemtosecond laser system may be helpful to collimate the fast electrons. This approachcan be relatively easily adopted in experiments and thus can find many interestingapplications when collimated fast electron transport is required.

Acknowledgments

This work was supported by the NNSFC (Grant Nos. 11775305, 11721091, 11774430,11665012, and 11475260), Science Challenge Project(Grant Nos. TZ2018001 andTZ2018005), and Open Fund of the State Key Laboratory of High Field Laser Physics(SIOM). X.H.Y. also acknowledges the support from the China Scholarship Council.

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